Elitza N. Maneva

Power-aware base station positioning for sensor networks

We consider the problem of positioning data collecting base stations in a sensor network. We show that in general, the choice of positions has a marked influence on the data rate, or equivalently, the power efficiency, of the network. In our model, which is partly motivated by an experimental environmental monitoring system, the optimum data rate for a fixed layout of base stations can be found by a maximum flow algorithm. Finding the optimum layout of base stations, however, turns out to be an NP-complete problem, even in the special case of homogeneous networks. Our analysis of the optimum layout for the special case of the regular grid shows that all layouts that meet certain constraints are equally good. We also consider two classes of random graphs, chosen to model networks that might be realistically encountered, and empirically evaluate the performance of several base station positioning algorithms on instances of these classes. In comparison to manually choosing positions along the periphery of the network or randomly choosing them within the network, the algorithms tested find positions, which significantly improve the data rate and power efficiency of the network.

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics, and threshold phenomena. Recent work on heuristics and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefers framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and

Lossy source encoding via message-passing and decimation over generalized codewords of LDGM codes

st

New model for rigorous analysis of LT-codes

-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side—which includes but is not limited to all problems with polynomial-time algorithms for satisfiability—is in P for the

Lossy Source Compression Using Low-Density Generator Matrix Codes: Analysis and Algorithms

st

Sherali--Adams Relaxations and Indistinguishability in Counting Logics

-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, diameter and complexity of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.

The connectivity of boolean satisfiability: computational and structural dichotomies

We describe message-passing and decimation approaches for lossy source coding using low-density generator matrix (LDGM) codes. In particular, this paper addresses the problem of encoding a Bernoulli(1/2) source: for randomly generated LDGM codes with suitably irregular degree distributions, our methods yield performance very close to the rate distortion limit over a range of rates. Our approach is inspired by the survey propagation (SP) algorithm, originally developed by Mezard et al. (2002) for solving random satisfiability problems. Previous work by Maneva et al. (2005) shows how SP can be understood as belief propagation (BP) for an alternative representation of satisfiability problems. In analogy to this connection, our approach is to define a family of Markov random fields over generalized codewords, from which local message-passing rules can be derived in the standard way. The overall source encoding method is based on message-passing, setting a subset of bits to their preferred values (decimation), and reducing the code

Pruning processes and a new characterization of convex geometries

We present a new model for LT codes which simplifies the analysis of the error probability of decoding by belief propagation. For any given degree distribution, we provide the first rigorous expression for the limiting bit-error probability as the length of the code goes to infinity via recent results in random hypergraphs by Darling and Norris, Ann. Appl. Probab., 2005. For a code of finite length, we provide an algorithm for computing the probability of block-error of the decoder. This algorithm improves by a linear factor the algorithm of Karp, Luby, and Shokrollahi, Proc. of ISIT, 2004

Sherali-Adams relaxations and indistinguishability in counting logics

We study the use of low-density generator matrix (LDGM) codes for lossy compression of the Bernoulli symmetric source. First, we establish rigorous upper bounds on the average distortion achieved by check-regular ensemble of LDGM codes under optimal minimum distance source encoding. These bounds establish that the average distortion using such bounded degree families rapidly approaches the Shannon limit as the degrees are increased. Second, we propose a family of message-passing algorithms, ranging from the standard belief propagation algorithm at one extreme to a variant of survey propagation algorithm at the other. When combined with a decimation subroutine and applied to LDGM codes with suitably irregular degree distributions, we show that such a message-passing/decimation algorithm yields distortion very close to the Shannon rate-distortion bound for the binary symmetric source.

Course 15 A hike in the phases of the 1-in-3 satisfiability

Two graphs with adjacency matrices